| Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology 1998 Edition Contributor(s): Spring, David (Editor) |
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ISBN: 376435805X ISBN-13: 9783764358051 Publisher: Birkhauser OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: December 1997 |
| Additional Information |
| BISAC Categories: - Mathematics | Topology - General |
| Dewey: 514.72 |
| LCCN: 97046762 |
| Series: Progress in Mathematics |
| Physical Information: 0.56" H x 6.69" W x 9.61" (1.24 lbs) 213 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 1. Historical Remarks Convex Integration theory, first introduced by M. Gromov 17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg 8]; (ii) the covering homotopy method which, following M. Gromov's thesis 16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale 36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par- tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov 18]). No such results on closed relations in jet spaees can be proved by means of the other two methods. |